Solve the inequality (x^2 - 5x + 6) / (x - 1) ≥ 0. Which of the following is the solution set?

Correct answer

B. (1, 2] ∪ [3, ∞)

Principle or equation

For a rational inequality, find critical points where numerator or denominator is zero, then test intervals. The denominator cannot be zero.

Why this answer is correct

Factor numerator: (x-2)(x-3). Critical points: x=1 (denominator zero), x=2, x=3. Test intervals: (-∞,1): numerator positive, denominator negative -> negative; (1,2]: numerator positive (or zero at 2), denominator positive -> nonnegative; [2,3]: numerator negative (or zero at endpoints), denominator positive -> nonpositive; [3,∞): numerator positive, denominator positive -> nonnegative. Exclude x=1 because denominator zero. So solution: (1,2] ∪ [3,∞).

Example

Solve (x-1)/(x-2) ≥ 0: critical points 1 and 2; solution (-∞,1] ∪ (2,∞).